or usehttp://www.ccl.net/chemistry/sub_unsub.shtmlhttp://www.ccl.net/spammers.txt--_000_13a6a065638c40f59e9e9ee2a1b4e1c2chtumde_ Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Dear Tom,

thanks for this perspective. It's interesting to differentiate the role = of charge partitioning schemes into (a) situations where you are trying to = efficiently expand a operator or electrostatic potential and (b) situations= where you are using the charges to interpret some chemical bonding situation.

In my personal opinion case (a) is far more important and interesting. A= s long as we are talking about population analysis in general (without spec= ifying the application), a statement like 'Mulliken is useless' (I'm paraph= rasing) is thus not helpful. However, since the original question on the list was about und= erstanding bonding based on population analysis, Mulliken is probably indee= d not the best choice here.

Best regards,

Hannes

From: owner-chemistry+j= ohannes.margraf=3D=3Dtum.de.:.ccl.net <owner-chemistry+johannes.margra= f=3D=3Dtum.de.:.ccl.net> on behalf of Thomas Manz thomasamanz%gmail.com <owner-chemistry.:.ccl.net>
Sent: Monday, June 29, 2020 5:54:12 PM
To: Margraf, Johannes
Subject: CCL: Charge

Hi Stefan,

I wanted to further clarify one aspect of my earlier response.

Suppose that one has a NaCl crystal, for example. Using a Mulliken pop= ulation analysis, depending on the basis set, the populations of the Na and= Cl atoms could take on any values that sum to 11 + 17 =3D 2= 8 electrons. For example, one could have 11.3488 electrons on the Na atom and 28 - 11.3488 =3D 16.6512 el= ectrons on the Cl atom. Using a different basis set, the Mulliken populatio= n analysis might yield 10.2342 electrons on the Na atom and 28 - 10.2342 = =3D 17.7658 electrons on the Cl atom. Using another basis set, you might get 11.0000 electrons on the Na atom and 28 - 11.0000 =3D 1= 7.0000 electrons on the Cl atom. For any chosen basis set, the Mulliken pop= ulations (or any other kind of populations) and their corresponding atomic = multipoles and charge penetration (i.e., Coulombic electrostatic interaction between overlapping functions) = integrals could be used as a representation to expand the Coulomb operator = (i.e., to calculate the Coulomb interaction between electrons in the q= uantum chemistry calculation). If the expansion is carried out to high enough order, then its precision could be= arbitrarily high (e.g., reproduce the Coulomb interaction to machine preci= sion). It is very clear to see this has nothing to do with chemically meani= ngful atom-in-material descriptors, because in case 1 the Na atom in the NaCl crystal would be assigned [sic] = as an anion and the Cl atom as a cation; in case 2 the Na atom in the NaCl = crystal would be assigned [sic] as an cation and the Cl atom as a anion; an= d in case 3 the Na atom in the NaCl crystal would be assigned [sic] as neutral (i.e., bearing no net charge) a= nd the Cl atom also as neutral. Hence, the Mulliken populations cannot be n= et atomic charges in general.

Sincerely,

Tom

On Mon, Jun 29, 2020 at 9:27 AM Thoma= s Manz <thomasamanz ~~ gmail= .com> wrote:

Hi Stefan,

I think the confusion arises, because the Mulliken populations are som= etimes confused with net atomic charges.

The expansion of the electron density can be performed using any desir= ed basis. In your application, the Mulliken partitioning is just a basis re= presentation for expanding the electron density in terms of a distribu= ted multipole expansion (e.g., up to quadrupole order). Yes, the Mulliken partitioning can be a mechanism to fo= rmulate a distributed multipole expansion of the electron density which can= be a useful computational algorithm for computing electrostatic interactio= ns during a quantum chemistry calculation. This is somewhat related to the fast multipole moments expansion of the Co= ulomb operator in quantum chemistry calculations. But, this is an entirely = different topic than extracting chemically meaningful atom-in-material desc= riptors from a quantum chemistry calculation.

Extracting chemically meaningful atom-in-material descriptors (net ato= mic charges, atomic spin moments, bond orders, s-p-d-f-g populations, etc.)= carries with it the extra requirements of exhibiting correlations to exper= imental observables and of having well-defined mathematical values (including a complete basis set limit) an= d of exhibiting chemical consistency between various chemical descriptors. = A multipole expansion of the Coulomb operator (such as the Mulliken-based m= ultipole expansion you mentioned) has nothing to do with chemically meaningful descriptors, it is simply a t= rick to re-write the density matrix using a different basis representation = to simplify the calculation of Coulomb integrals. In other words, it is mer= ely algorithmic.

The great confusion regarding Mulliken populations, which are simply m= athematical artifices and not chemical properties, is that they have histor= ically been confused with chemical properties like net atomic charges. Just= like basis set overlap integrals, Mulliken populations can be a useful ingredient for expanding the Coulomb = operator, as your example illustrates, but they are no more chemical proper= ties of a material than basis set overlap integrals are chemical properties= of material. In other words, not everything used in a quantum chemistry calculation is a chemical property = of a material: some are just mathematical constructs whose utility resides = in the algorithmic computation of another quantity (e.g., electrostatic int= eraction). The origin of this great confusion is that for small basis sets the Mulliken populations often rese= mble the net atomic charges computed by other methods, but this is somewhat= coincidental because the correlation fails to hold when the basis set is i= mproved.

The reason this often confuses people is because there are actually tw= o opposite ways to construct a polyatomic multipole expansion:

(a) using quantities that are merely algorithmic (e.g., Mulliken popul= ations) in the sense they have no complete basis set limit but none-the-les= s can be used as a basis representation to expand the Coulomb potential and=

(b) using chemically well-defined quantities (e.g., DDEC6 or QTAIM or = Hirshfeld NACs and atomic multipoles) that have well-defined complete basis= set limits and can be used as a basis representation to expand the Coulomb= potential

People often fail to recognize the distinction between these two cases= , which have a day and night difference between them.

Sincerely,

Tom

On Mon, Jun 29, 2020 at 8:43 AM Stefa= n Grimme grimme**thch= .uni-bonn.de <owner-chemistry ~~ ccl.net> wrote:

Sent to CCL by: "Stefan Grimme" [grimme..thch.uni-bonn.de= ]
One more comment to the Mulliken charge discussion:
even methods without a well-defined basis set limit can be useful
as already mentioned by Marcel Swart. This holds
for the Mulliken atomic charge partitioning in compact MB/DZ basis sets
(even TZ is often reasonable). For example the DFTB and GFN-xTB tight-bindi= ng methods are fundamentally based on a Mulliken analysis of the density ma= trix and yield physically very reasonable electrostatic energies. In GFN2-x= TB this also works well up to quadrupole moments.
Its clear that the Mulliken scheme breaks down for AO basis sets containing=
diffuse components but I really would like to see a differentiated view on = the topic (and not as in a recent general statement of a reviewer something= like "I do not think Mulliken charges are trustworthy").
Best
Stefan Grimme

-=3D This is automatically added to each message by the mailing script =3D-=
E-mail to subscribers: CHEMISTRY ~~ ccl.net or use:

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--_000_13a6a065638c40f59e9e9ee2a1b4e1c2chtumde_-- From owner-chemistry@ccl.net Tue Jun 30 11:38:01 2020 From: "Thomas Manz thomasamanz-x-gmail.com" To: CCL Subject: CCL: Charge Message-Id: <-54126-200630113357-17227-95JepoXYlM8UaMVAoewMww##server.ccl.net> X-Original-From: Thomas Manz Content-Type: multipart/alternative; boundary="000000000000d293a305a94ee61f" Date: Tue, 30 Jun 2020 09:33:37 -0600 MIME-Version: 1.0 Sent to CCL by: Thomas Manz [thomasamanz,,gmail.com] --000000000000d293a305a94ee61f Content-Type: text/plain; charset="UTF-8" Hi Hannes, This is not a green light that Mulliken population analysis is physically meaningful. It just means that it may be useful for computing other things that are physically meaningful like expanding the electrostatic potential or localizing orbitals (e.g., via the Pipek-Mezey scheme). For example, one could use a Mulliken-like population analysis method to localize orbitals for the purpose of achieving a linear-scaling coupled-cluster algorithm. Some approaches in fact do this or something extremely similar. The reason such can be done is this expansion improves the computational efficiency but does not alter the numeric result. One could even argue for truncations of such expansions that change the numeric results slightly, but not too much. Whenever one discusses the specific computed values of atom-in-material descriptors, whether bond orders, s-p-d-f-g subshell populations, partial charges, atomic spin moments, specifica bond order components, etc. in scientific publications, those values should be computed by a population analysis method having a complete basis set limit. (The chosen population analysis method should also meet the criteria of being highly correlated to many experimentally measured properties (e.g., spectroscopic results for carefully chosen benchmark systems) and also yield various atom-in-material descriptors that are chemically consistent among themselves.) There is rarely a good reason to discuss specific Mulliken population values in scientific publications anymore. Those were useful long ago when Mulliken population analysis was first introduced when there were no good atomic population analysis methods available. In recent decades, there has been a sort of "free for all" where many extremely undisciplined and poor performing atomic population analysis methods were introduced, often with inadequate testing and over-inflated claims. A small number of high-quality atomic population analysis methods were also introduced over the past few decades. The quantum chemistry field and more generally all chemical sciences are now facing "growing pains" where they need to transition to a future that includes extremely well-constructed atom-in-material descriptors. As with nearly all major transitions, certain aspects of this transformation will not be easy, such as the need to retool software packages to facilitate calculations using the best available methods. Textbooks will also need substantially updated, which will take some time. In the end, the effort will be worth it. Sincerely, Tom On Tue, Jun 30, 2020 at 3:43 AM Margraf, Johannes johannes.margraf## ch.tum.de wrote: > Dear Tom, > > > thanks for this perspective. It's interesting to differentiate the role of > charge partitioning schemes into (a) situations where you are trying to > efficiently expand a operator or electrostatic potential and (b) situations > where you are using the charges to interpret some chemical bonding > situation. > > > In my personal opinion case (a) is far more important and interesting. As > long as we are talking about population analysis in general (without > specifying the application), a statement like 'Mulliken is useless' (I'm > paraphrasing) is thus not helpful. However, since the original question on > the list was about understanding bonding based on population analysis, > Mulliken is probably indeed not the best choice here. > > > Best regards, > > > Hannes > > ________________________________ > > From: owner-chemistry+johannes.margraf==tum.de(_)ccl.net > on behalf of Thomas > Manz thomasamanz%gmail.com > Sent: Monday, June 29, 2020 5:54:12 PM > To: Margraf, Johannes > Subject: CCL: Charge > > Hi Stefan, > > I wanted to further clarify one aspect of my earlier response. > > Suppose that one has a NaCl crystal, for example. Using a Mulliken > population analysis, depending on the basis set, the populations of the Na > and Cl atoms could take on any values that sum to 11 + 17 = 28 electrons. > For example, one could have 11.3488 electrons on the Na atom and 28 - > 11.3488 = 16.6512 electrons on the Cl atom. Using a different basis set, > the Mulliken population analysis might yield 10.2342 electrons on the Na > atom and 28 - 10.2342 = 17.7658 electrons on the Cl atom. Using another > basis set, you might get 11.0000 electrons on the Na atom and 28 - 11.0000 > = 17.0000 electrons on the Cl atom. For any chosen basis set, the Mulliken > populations (or any other kind of populations) and their corresponding > atomic multipoles and charge penetration (i.e., Coulombic electrostatic > interaction between overlapping functions) integrals could be used as a > representation to expand the Coulomb operator (i.e., to calculate the > Coulomb interaction between electrons in the quantum chemistry > calculation). If the expansion is carried out to high enough order, then > its precision could be arbitrarily high (e.g., reproduce the Coulomb > interaction to machine precision). It is very clear to see this has nothing > to do with chemically meaningful atom-in-material descriptors, because in > case 1 the Na atom in the NaCl crystal would be assigned [sic] as an anion > and the Cl atom as a cation; in case 2 the Na atom in the NaCl crystal > would be assigned [sic] as an cation and the Cl atom as a anion; and in > case 3 the Na atom in the NaCl crystal would be assigned [sic] as neutral > (i.e., bearing no net charge) and the Cl atom also as neutral. Hence, the > Mulliken populations cannot be net atomic charges in general. > > Sincerely, > > Tom > > On Mon, Jun 29, 2020 at 9:27 AM Thomas Manz > wrote: > Hi Stefan, > > I think the confusion arises, because the Mulliken populations are > sometimes confused with net atomic charges. > > The expansion of the electron density can be performed using any desired > basis. In your application, the Mulliken partitioning is just a basis > representation for expanding the electron density in terms of a distributed > multipole expansion (e.g., up to quadrupole order). Yes, the Mulliken > partitioning can be a mechanism to formulate a distributed multipole > expansion of the electron density which can be a useful computational > algorithm for computing electrostatic interactions during a quantum > chemistry calculation. This is somewhat related to the fast multipole > moments expansion of the Coulomb operator in quantum chemistry > calculations. But, this is an entirely different topic than extracting > chemically meaningful atom-in-material descriptors from a quantum chemistry > calculation. > > Extracting chemically meaningful atom-in-material descriptors (net atomic > charges, atomic spin moments, bond orders, s-p-d-f-g populations, etc.) > carries with it the extra requirements of exhibiting correlations to > experimental observables and of having well-defined mathematical values > (including a complete basis set limit) and of exhibiting chemical > consistency between various chemical descriptors. A multipole expansion of > the Coulomb operator (such as the Mulliken-based multipole expansion you > mentioned) has nothing to do with chemically meaningful descriptors, it is > simply a trick to re-write the density matrix using a different basis > representation to simplify the calculation of Coulomb integrals. In other > words, it is merely algorithmic. > > The great confusion regarding Mulliken populations, which are simply > mathematical artifices and not chemical properties, is that they have > historically been confused with chemical properties like net atomic > charges. Just like basis set overlap integrals, Mulliken populations can be > a useful ingredient for expanding the Coulomb operator, as your example > illustrates, but they are no more chemical properties of a material than > basis set overlap integrals are chemical properties of material. In other > words, not everything used in a quantum chemistry calculation is a chemical > property of a material: some are just mathematical constructs whose utility > resides in the algorithmic computation of another quantity (e.g., > electrostatic interaction). The origin of this great confusion is that for > small basis sets the Mulliken populations often resemble the net atomic > charges computed by other methods, but this is somewhat coincidental > because the correlation fails to hold when the basis set is improved. > > The reason this often confuses people is because there are actually two > opposite ways to construct a polyatomic multipole expansion: > > (a) using quantities that are merely algorithmic (e.g., Mulliken > populations) in the sense they have no complete basis set limit but > none-the-less can be used as a basis representation to expand the Coulomb > potential and > > (b) using chemically well-defined quantities (e.g., DDEC6 or QTAIM or > Hirshfeld NACs and atomic multipoles) that have well-defined complete basis > set limits and can be used as a basis representation to expand the Coulomb > potential > > People often fail to recognize the distinction between these two cases, > which have a day and night difference between them. > > Sincerely, > > Tom > > > On Mon, Jun 29, 2020 at 8:43 AM Stefan Grimme grimme**thch.uni-bonn.de< > http://thch.uni-bonn.de> owner-chemistry%20~~%20ccl.net>> wrote: > > Sent to CCL by: "Stefan Grimme" [grimme..thch.uni-bonn.de< > http://thch.uni-bonn.de>] > One more comment to the Mulliken charge discussion: > even methods without a well-defined basis set limit can be useful > as already mentioned by Marcel Swart. This holds > for the Mulliken atomic charge partitioning in compact MB/DZ basis sets > (even TZ is often reasonable). For example the DFTB and GFN-xTB > tight-binding methods are fundamentally based on a Mulliken analysis of the > density matrix and yield physically very reasonable electrostatic energies. > In GFN2-xTB this also works well up to quadrupole moments. > Its clear that the Mulliken scheme breaks down for AO basis sets containing > diffuse components but I really would like to see a differentiated view on > the topic (and not as in a recent general statement of a reviewer something > like "I do not think Mulliken charges are trustworthy"). > Best > Stefan Grimme> E-mail to subscribers: CHEMISTRY ~~ ccl.net 20ccl.net> or use:E-mail to administrators: CHEMISTRY-REQUEST ~~ ccl.net > or usehttp:// > www.ccl.net/chemistry/sub_unsub.shtmlhttp://www.ccl.net/spammers.txt--_000_13a6a065638c40f59e9e9ee2a1b4e1c2chtumde_ > Content-Type > : > text/html; charset"us-ascii" > Content-Transfer-Encoding: quoted-printable > > > > > > > >

style="font-size:12pt;color:#000000;font-family:Calibri,Helvetica,sans-serif;" > dir="ltr"> >

Dear Tom,

thanks for this perspective. It's interesting to differentiate the role > of charge partitioning schemes into (a) situations where you are trying to > efficiently expand a operator or electrostatic potential and (b) situations > where you are using the charges > to interpret some chemical bonding situation.

In my personal opinion case (a) is far more important and interesting. > As long as we are talking about population analysis in general (without > specifying the application), a statement like 'Mulliken is useless' (I'm > paraphrasing) is thus not helpful. > However, since the original question on the list was about > understanding bonding based on population analysis, Mulliken is probably > indeed not the best choice here.

Best regards,

Hannes
>

style="font-size:11pt" color="#000000">From: > owner-chemistry+johannes.margraf==tum.de(_)ccl.net > <owner-chemistry+johannes.margraf==tum.de(_)ccl.net> on behalf > of Thomas Manz thomasamanz%gmail.com > <owner-chemistry(_)ccl.net>
> Sent: Monday, June 29, 2020 5:54:12 PM
> To: Margraf, Johannes
> Subject: CCL: Charge >

Hi Stefan, >

I wanted to further clarify one aspect of my earlier response.

Suppose that one has a NaCl crystal, for example. Using a Mulliken > population analysis, depending on the basis set, the populations of the Na > and Cl atoms could take on any values that sum to 11 + 17 = > 28 electrons. For example, one could have 11.3488 > electrons on the Na atom and 28 - 11.3488 = 16.6512 > electrons on the Cl atom. Using a different basis set, the Mulliken > population analysis might yield 10.2342 electrons on the Na atom and 28 - > 10.2342 = 17.7658 electrons on the Cl atom. Using another basis > set, you might get 11.0000 electrons on the Na atom and 28 - 11.0000 = > 17.0000 electrons on the Cl atom. For any chosen basis set, the Mulliken > populations (or any other kind of populations) and their corresponding > atomic multipoles and charge penetration > (i.e., Coulombic electrostatic interaction between overlapping functions) > integrals could be used as a representation to expand the Coulomb operator > (i.e., to calculate the Coulomb interaction between electrons in the > quantum chemistry calculation). If the > expansion is carried out to high enough order, then its precision could > be arbitrarily high (e.g., reproduce the Coulomb interaction to machine > precision). It is very clear to see this has nothing to do with chemically > meaningful atom-in-material descriptors, > because in case 1 the Na atom in the NaCl crystal would be assigned [sic] > as an anion and the Cl atom as a cation; in case 2 the Na atom in the NaCl > crystal would be assigned [sic] as an cation and the Cl atom as a anion; > and in case 3 the Na atom in the NaCl > crystal would be assigned [sic] as neutral (i.e., bearing no net charge) > and the Cl atom also as neutral. Hence, the Mulliken populations cannot be > net atomic charges in general.

Sincerely,

Tom

>
>

On Mon, Jun 29, 2020 at 9:27 AM Thomas > Manz <thomasamanz ~~ > gmail.com> wrote:
>

>
Hi Stefan, >

>
>
I think the confusion arises, because the Mulliken populations are > sometimes confused with net atomic charges.
>

>
>
The expansion of the electron density can be performed using any > desired basis. In your application, the Mulliken partitioning is just a > basis representation for expanding the electron density in terms of a > distributed multipole expansion (e.g., up to > quadrupole order). Yes, the Mulliken partitioning can be a mechanism to > formulate a distributed multipole expansion of the electron density which > can be a useful computational algorithm for computing electrostatic > interactions during a quantum chemistry calculation. > This is somewhat related to the fast multipole moments expansion of the > Coulomb operator in quantum chemistry calculations. But, this is an > entirely different topic than extracting chemically meaningful > atom-in-material descriptors from a quantum chemistry > calculation.
>

>
>
Extracting chemically meaningful atom-in-material descriptors (net > atomic charges, atomic spin moments, bond orders, s-p-d-f-g populations, > etc.) carries with it the extra requirements of exhibiting correlations to > experimental observables and of having > well-defined mathematical values (including a complete basis set limit) > and of exhibiting chemical consistency between various chemical > descriptors. A multipole expansion of the Coulomb operator (such as the > Mulliken-based multipole expansion you mentioned) > has nothing to do with chemically meaningful descriptors, it is simply a > trick to re-write the density matrix using a different basis representation > to simplify the calculation of Coulomb integrals. In other words, it is > merely algorithmic.
>

>
>
The great confusion regarding Mulliken populations, which are simply > mathematical artifices and not chemical properties, is that they have > historically been confused with chemical properties like net atomic > charges. Just like basis set overlap integrals, > Mulliken populations can be a useful ingredient for expanding the Coulomb > operator, as your example illustrates, but they are no more chemical > properties of a material than basis set overlap integrals are chemical > properties of material. In other words, not > everything used in a quantum chemistry calculation is a chemical property > of a material: some are just mathematical constructs whose utility resides > in the algorithmic computation of another quantity (e.g., electrostatic > interaction). The origin of this great > confusion is that for small basis sets the Mulliken populations often > resemble the net atomic charges computed by other methods, but this is > somewhat coincidental because the correlation fails to hold when the basis > set is improved.
>

>
>
The reason this often confuses people is because there are actually > two opposite ways to construct a polyatomic multipole expansion:
>

>
>
(a) using quantities that are merely algorithmic (e.g., Mulliken > populations) in the sense they have no complete basis set limit but > none-the-less can be used as a basis representation to expand the Coulomb > potential and
>

>
>
(b) using chemically well-defined quantities (e.g., DDEC6 or QTAIM or > Hirshfeld NACs and atomic multipoles) that have well-defined complete basis > set limits and can be used as a basis representation to expand the Coulomb > potential >
>

>
>
People often fail to recognize the distinction between these two > cases, which have a day and night difference between them.
>
>

>
>
Sincerely,
>

>
>
Tom
>

>
>
>
>
On Mon, Jun 29, 2020 at 8:43 AM Stefan > Grimme grimme** > thch.uni-bonn.de < target="_blank">owner-chemistry ~~ ccl.net> wrote:
>
>
>
> Sent to CCL by: "Stefan Grimme" [grimme..thch.uni-bonn.de > ]
> One more comment to the Mulliken charge discussion:
> even methods without a well-defined basis set limit can be useful
> as already mentioned by Marcel Swart. This holds
> for the Mulliken atomic charge partitioning in compact MB/DZ basis sets
> (even TZ is often reasonable). For example the DFTB and GFN-xTB > tight-binding methods are fundamentally based on a Mulliken analysis of the > density matrix and yield physically very reasonable electrostatic energies. > In GFN2-xTB this also works well up to quadrupole > moments.
> Its clear that the Mulliken scheme breaks down for AO basis sets > containing
> diffuse components but I really would like to see a differentiated view on > the topic (and not as in a recent general statement of a reviewer something > like "I do not think Mulliken charges are trustworthy").
> Best
> Stefan Grimme
>
>
>
> -= This is automatically added to each message by the mailing script > =- >
> E-mail to subscribers: target="_blank">CHEMISTRY ~~ ccl.net or use:
> target="_blank"
>
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> target="_blank"
> target="_blank"
>
> Before posting, check wait time at: rel="noreferrer" target="_blank"> > http://www.ccl.net
>
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>
> Search Messages: target="_blank"> > http://www.ccl.net/chemistry/searchccl/index.shtml
> rel="noreferrer" target="_blank"
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>
>
>
>
>

> > > --000000000000d293a305a94ee61f Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable

Hi Hannes,

This is not a gre= en light that Mulliken population analysis is physically meaningful. It jus= t means that it may be useful for computing other things that are physicall= y meaningful like expanding the electrostatic potential or localizing orbit= als (e.g., via the Pipek-Mezey scheme). For example, one could use a Mullik= en-like population analysis method to localize orbitals for the purpose of = achieving a linear-scaling coupled-cluster algorithm. Some approaches in fa= ct do this or something extremely similar. The reason such can be done is t= his expansion improves the computational efficiency but does not alter the = numeric result. One could even argue for truncations of such expansions tha= t change the numeric results slightly, but not too much.

Whenever one discusses the specific computed values of atom-in-m= aterial descriptors, whether bond orders, s-p-d-f-g subshell populations, p= artial charges, atomic spin moments, specifica bond order components, etc. = in scientific publications, those values should be computed by a population= analysis method having a complete basis set limit. (The chosen population = analysis method should also meet the criteria of being highly correlated to= many experimentally measured properties (e.g., spectroscopic results for c= arefully chosen benchmark systems) and also yield various atom-in-material = descriptors that are chemically consistent among themselves.) There is rare= ly a good reason to discuss specific Mulliken population values in scientif= ic publications anymore. Those were useful long ago when Mulliken populatio= n analysis was first introduced when there were no good atomic=C2=A0populat= ion=C2=A0analysis=C2=A0methods available.=C2=A0

In= recent decades, there has been a sort of "free for all" where ma= ny extremely undisciplined and poor performing atomic population analysis m= ethods were introduced, often with inadequate testing and over-inflated cla= ims. A small number of high-quality atomic population analysis methods were= also introduced over the past few decades. The quantum chemistry field and= more generally all chemical sciences are now facing "growing pains&qu= ot; where they need to transition to a future that includes extremely well-= constructed atom-in-material descriptors. As with nearly all major transiti= ons, certain aspects of this transformation will not be easy, such as the n= eed to retool software packages to facilitate calculations using the best a= vailable methods. Textbooks will also need substantially updated, which wil= l take some time. In the end, the effort will be worth it.=C2=A0

Sincerely,

Tom

On Tue, Jun 30, 2020= at 3:43 AM Margraf, Johannes johannes.margraf##ch.tum.de <owner-chemis= try/./ccl.net> wrote:

Dear Tom,

thanks for this perspective. It's interesting to differentiate the role= of charge partitioning schemes into (a) situations where you are trying to= efficiently expand a operator or electrostatic potential and (b) situation= s where you are using the charges to interpret some chemical bonding situat= ion.

In my personal opinion case (a) is far more important and interesting. As l= ong as we are talking about population analysis in general (without specify= ing the application), a statement like 'Mulliken is useless' (I'= ;m paraphrasing) is thus not helpful. However, since the original question = on the list was about understanding bonding based on population analysis, M= ulliken is probably indeed not the best choice here.

Best regards,

Hannes

________________________________
> From: owner-chemistry+johannes.margraf=3D=3Dtum.de(_)ccl.net <owner-chemistry+joha= nnes.margraf=3D=3Dtum.de(_)ccl.net> on behalf of Thomas Manz thomasamanz%gmail.com <owner= -chemistry(_)ccl.net>
Sent: Monday, June 29, 2020 5:54:12 PM
To: Margraf, Johannes
Subject: CCL: Charge

Hi Stefan,

I wanted to further clarify one aspect of my earlier response.

Suppose that one has a NaCl crystal, for example. Using a Mulliken populati= on analysis, depending on the basis set, the populations of the Na and Cl a= toms could take on any values that sum to 11 + 17=C2=A0 =3D 28 electrons. F= or example, one could have 11.3488 electrons on the Na atom and 28=C2=A0 -= =C2=A0 11.3488 =3D=C2=A0 16.6512 electrons on the Cl atom. Using a differen= t basis set, the Mulliken population analysis might yield 10.2342 electrons= on the Na atom and 28 - 10.2342 =3D 17.7658 electrons on the Cl atom. Usin= g another basis set, you might get 11.0000 electrons on the Na atom and 28 = - 11.0000 =3D 17.0000 electrons on the Cl atom. For any chosen basis set, t= he Mulliken populations (or any other kind of populations) and their corres= ponding atomic multipoles and charge penetration (i.e., Coulombic electrost= atic interaction between overlapping functions) integrals could be used as = a representation to expand the Coulomb operator (i.e., to calculate the Cou= lomb interaction between electrons in the quantum chemistry calculation). I= f the expansion is carried out to high enough order, then its precision cou= ld be arbitrarily high (e.g., reproduce the Coulomb interaction to machine = precision). It is very clear to see this has nothing to do with chemically = meaningful atom-in-material descriptors, because in case 1 the Na atom in t= he NaCl crystal would be assigned [sic] as an anion and the Cl atom as a ca= tion; in case 2 the Na atom in the NaCl crystal would be assigned [sic] as = an cation and the Cl atom as a anion; and in case 3 the Na atom in the NaCl= crystal would be assigned [sic] as neutral (i.e., bearing no net charge) a= nd the Cl atom also as neutral. Hence, the Mulliken populations cannot be n= et atomic charges in general.

Sincerely,

Tom

On Mon, Jun 29, 2020 at 9:27 AM Thomas Manz <thomasamanz ~~ gmail.com<mail= to:thomasamanz%20= ~~%20gm= ail.com>> wrote:
Hi Stefan,

I think the confusion arises, because the Mulliken populations are sometime= s confused with net atomic charges.

The expansion of the electron density can be performed using any desired ba= sis. In your application, the Mulliken partitioning is just a basis represe= ntation for expanding the electron density in terms of a distributed multip= ole expansion (e.g., up to quadrupole order). Yes, the Mulliken partitionin= g can be a mechanism to formulate a distributed multipole expansion of the = electron density which can be a useful computational algorithm for computin= g electrostatic interactions during a quantum chemistry calculation. This i= s somewhat related to the fast multipole moments expansion of the Coulomb o= perator in quantum chemistry calculations. But, this is an entirely differe= nt topic than extracting chemically meaningful atom-in-material descriptors= from a quantum chemistry calculation.

Extracting chemically meaningful atom-in-material descriptors (net atomic c= harges, atomic spin moments, bond orders, s-p-d-f-g populations, etc.) carr= ies with it the extra requirements of exhibiting correlations to experiment= al observables and of having well-defined mathematical values (including a = complete basis set limit) and of exhibiting chemical consistency between va= rious chemical descriptors. A multipole expansion of the Coulomb operator (= such as the Mulliken-based multipole expansion you mentioned) has nothing t= o do with chemically meaningful descriptors, it is simply a trick to re-wri= te the density matrix using a different basis representation to simplify th= e calculation of Coulomb integrals. In other words, it is merely algorithmi= c.

The great confusion regarding Mulliken populations, which are simply mathem= atical artifices and not chemical properties, is that they have historicall= y been confused with chemical properties like net atomic charges. Just like= basis set overlap integrals, Mulliken populations can be a useful ingredie= nt for expanding the Coulomb operator, as your example illustrates, but the= y are no more chemical properties of a material than basis set overlap inte= grals are chemical properties of material. In other words, not everything u= sed in a quantum chemistry calculation is a chemical property of a material= : some are just mathematical constructs whose utility resides in the algori= thmic computation of another quantity (e.g., electrostatic interaction). Th= e origin of this great confusion is that for small basis sets the Mulliken = populations often resemble the net atomic charges computed by other methods= , but this is somewhat coincidental because the correlation fails to hold w= hen the basis set is improved.

The reason this often confuses people is because there are actually two opp= osite ways to construct a polyatomic multipole expansion:

(a) using quantities that are merely algorithmic (e.g., Mulliken population= s) in the sense they have no complete basis set limit but none-the-less can= be used as a basis representation to expand the Coulomb potential and

(b) using chemically well-defined quantities (e.g., DDEC6 or QTAIM or Hirsh= feld NACs and atomic multipoles) that have well-defined complete basis set = limits and can be used as a basis representation to expand the Coulomb pote= ntial

People often fail to recognize the distinction between these two cases, whi= ch have a day and night difference between them.

Sincerely,

Tom

On Mon, Jun 29, 2020 at 8:43 AM Stefan Grimme grimme**thch.uni-bonn.de<= ;h= ttp://thch.uni-bonn.de> <owner-chemistry ~~ ccl.net<mailto:owner-chemistry%20~~%20ccl.net>> wrote:

Sent to CCL by: "Stefan=C2=A0 Grimme" [grimme..thch.uni-bonn.de= <http://thch.uni-bonn.de>]
One more comment to the Mulliken charge discussion:
even methods without a well-defined basis set limit can be useful
as already mentioned by Marcel Swart. This holds
for the Mulliken atomic charge partitioning in compact MB/DZ basis sets
(even TZ is often reasonable). For example the DFTB and GFN-xTB tight-bindi= ng methods are fundamentally based on a Mulliken analysis of the density ma= trix and yield physically very reasonable electrostatic energies. In GFN2-x= TB this also works well up to quadrupole moments.
Its clear that the Mulliken scheme breaks down for AO basis sets containing=
diffuse components but I really would like to see a differentiated view on = the topic (and not as in a recent general statement of a reviewer something= like "I do not think Mulliken charges are trustworthy").
Best
Stefan Grimme

-=3D This is automatically added to each message by the mailing script =3D-=
E-mail to subscribers: CHEMISTRY ~~ ccl.net<mailto:CHEMISTRY%20~~%20ccl.net> or use:E-mail to admini= strators: CHEMISTRY-REQUEST ~~ ccl.net<mailto:CHEMISTRY-REQUEST%20~~%20ccl.net> or usehttp:/= /www.ccl.net/chemistry/sub_unsub.shtmlhtt= p://www.ccl.net/spammers.txt--_000_13a6a065638c40f59e9e9ee2a1b4e1c2chtumde_=
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<div id=3D"divtagdefaultwrapper" style=3D"font-size:12pt;= color:#000000;font-family:Calibri,Helvetica,sans-serif;" dir=3D"l= tr">
<p>Dear Tom,</p>
<p><br>
</p>
<p>thanks for this perspective. It's interesting to differentiate= the role of charge partitioning schemes into (a) situations where you are = trying to efficiently expand a operator or electrostatic potential and (b) = situations where you are using the charges
=C2=A0to interpret some chemical bonding situation.</p>
<p><br>
</p>
<p>In my personal opinion case (a) is far more important and interest= ing. As long as we are talking about population analysis in general (withou= t specifying the application), a statement like 'Mulliken is useless= 9; (I'm paraphrasing) is thus not helpful.
<span>However</span>, since the original question on the list w= as about understanding bonding based on population analysis, Mulliken is pr= obably indeed not the best choice here.</p>
<p><br>
</p>
<p>Best regards,</p>
<p><br>
</p>
<p>Hannes<br>
</p>
</div>
<hr style=3D"display:inline-block;width:98%" tabindex=3D"= -1">
<div id=3D"divRplyFwdMsg" dir=3D"ltr"><font fa= ce=3D"Calibri, sans-serif" style=3D"font-size:11pt" col= or=3D"#000000"><b>From:</b> owner-chemistry&#4= 3;johannes.margraf=3D=3Dtum.de(_)ccl.net <owner-chemistry+johannes.margraf= =3D=3Dtum.de= (_)ccl.= net> on behalf of Thomas Manz thomasamanz%gmail.com
=C2=A0<owner-chemistry(_)ccl.net><br>
<b>Sent:</b> Monday, June 29, 2020 5:54:12 PM<br>
<b>To:</b> Margraf, Johannes<br>
<b>Subject:</b> CCL: Charge</font>
<div> </div>
</div>
<div>
<div dir=3D"ltr">Hi Stefan,
<div><br>
</div>
<div>I wanted to further clarify one aspect of my earlier response.&l= t;/div>
<div><br>
</div>
<div>Suppose that one has a NaCl crystal, for example. Using a Mullik= en population analysis, depending on the basis set, the populations of the = Na and Cl atoms could take on any values that sum to 11 + = 17  =3D 28 electrons. For example, one could have 11.3488
=C2=A0electrons on the Na atom and 28  -  11.3488 =3D&= ;nbsp; 16.6512 electrons on the Cl atom. Using a different basis set, the M= ulliken population analysis might yield 10.2342 electrons on the Na atom an= d 28 - 10.2342 =3D 17.7658 electrons on the Cl atom. Using another basis =C2=A0set, you might get 11.0000 electrons on the Na atom and 28 - 11.0000 = =3D 17.0000 electrons on the Cl atom. For any chosen basis set, the Mullike= n populations (or any other kind of populations) and their corresponding at= omic multipoles and charge penetration
=C2=A0(i.e., Coulombic electrostatic interaction between overlapping functi= ons) integrals could be used as a representation to expand the Coulomb oper= ator (i.e., to calculate the Coulomb interaction between electrons= in the quantum chemistry calculation). If the
=C2=A0expansion is carried out to high enough order, then its precision cou= ld be arbitrarily high (e.g., reproduce the Coulomb interaction to machine = precision). It is very clear to see this has nothing to do with chemically = meaningful atom-in-material descriptors,
=C2=A0because in case 1 the Na atom in the NaCl crystal would be assigned [= sic] as an anion and the Cl atom as a cation; in case 2 the Na atom in the = NaCl crystal would be assigned [sic] as an cation and the Cl atom as a anio= n; and in case 3 the Na atom in the NaCl
=C2=A0crystal would be assigned [sic] as neutral (i.e., bearing no net char= ge) and the Cl atom also as neutral. Hence, the Mulliken populations cannot= be net atomic charges in general.</div>
<div><br>
</div>
<div>Sincerely,</div>
<div><br>
</div>
<div>Tom</div>
</div>
<br>
<div class=3D"gmail_quote">
<div dir=3D"ltr" class=3D"gmail_attr">On Mon, Jun= 29, 2020 at 9:27 AM Thomas Manz <<a href=3D"mailto:thomasamanz ~~ gmail.com">= thomasamanz ~~ gmail.com</a>> wrote:<br>
</div>
<blockquote class=3D"gmail_quote" style=3D"margin:0px 0px= 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">= ;
<div dir=3D"ltr">Hi Stefan,
<div><br>
</div>
<div>I think the confusion arises, because the Mulliken populations a= re sometimes confused with net atomic charges.</div>
<div><br>
</div>
<div>The expansion of the electron density can be performed using any= desired basis. In your application, the Mulliken partitioning is just a ba= sis representation for expanding the electron density in terms of = a distributed multipole expansion (e.g., up to
=C2=A0quadrupole order). Yes, the Mulliken partitioning can be a mechanism = to formulate a distributed multipole expansion of the electron density whic= h can be a useful computational algorithm for computing electrostatic inter= actions during a quantum chemistry calculation.
=C2=A0This is somewhat related to the fast multipole moments expansion of t= he Coulomb operator in quantum chemistry calculations. But, this is an enti= rely different topic than extracting chemically meaningful atom-in-material= descriptors from a quantum chemistry
=C2=A0calculation.</div>
<div><br>
</div>
<div>Extracting chemically meaningful atom-in-material descriptors (n= et atomic charges, atomic spin moments, bond orders, s-p-d-f-g populations,= etc.) carries with it the extra requirements of exhibiting correlations to= experimental observables and of having
=C2=A0well-defined mathematical values (including a complete basis set limi= t) and of exhibiting chemical consistency between various chemical descript= ors. A multipole expansion of the Coulomb operator (such as the Mulliken-ba= sed multipole expansion you mentioned)
=C2=A0has nothing to do with chemically meaningful descriptors, it is simpl= y a trick to re-write the density matrix using a different basis representa= tion to simplify the calculation of Coulomb integrals. In other words, it i= s merely algorithmic.</div>
<div><br>
</div>
<div>The great confusion regarding Mulliken populations, which are si= mply mathematical artifices and not chemical properties, is that they have = historically been confused with chemical properties like net atomic charges= . Just like basis set overlap integrals,
=C2=A0Mulliken populations can be a useful ingredient for expanding the Cou= lomb operator, as your example illustrates, but they are no more chemical p= roperties of a material than basis set overlap integrals are chemical prope= rties of material. In other words, not
=C2=A0everything used in a quantum chemistry calculation is a chemical prop= erty of a material: some are just mathematical constructs whose utility res= ides in the algorithmic computation of another quantity (e.g., electrostati= c interaction). The origin of this great
=C2=A0confusion is that for small basis sets the Mulliken populations often= resemble the net atomic charges computed by other methods, but this is som= ewhat coincidental because the correlation fails to hold when the basis set= is improved.</div>
<div><br>
</div>
<div>The reason this often confuses people is because there are actua= lly two opposite ways to construct a polyatomic multipole expansion:&nb= sp;</div>
<div><br>
</div>
<div>(a) using quantities that are merely algorithmic (e.g., Mulliken= populations) in the sense they have no complete basis set limit but none-t= he-less can be used as a basis representation to expand the Coulomb potenti= al and </div>
<div><br>
</div>
<div>(b) using chemically well-defined quantities (e.g., DDEC6 or QTA= IM or Hirshfeld NACs and atomic multipoles) that have well-defined complete= basis set limits and can be used as a basis representation to expand the C= oulomb potential
</div>
<div><br>
</div>
<div>People often fail to recognize the distinction between these two= cases, which have a day and night difference between them.<br>
</div>
<div><br>
</div>
<div>Sincerely,</div>
<div><br>
</div>
<div>Tom</div>
<div> </div>
</div>
<br>
<div class=3D"gmail_quote">
<div dir=3D"ltr" class=3D"gmail_attr">On Mon, Jun= 29, 2020 at 8:43 AM Stefan Grimme grimme**<a href=3D"http://thch.uni= -bonn.de" target=3D"_blank">thch.uni-bonn.de</a&= gt; <<a href=3D"mailto:owner-chemistry ~~ ccl.net" target=3D"_blank"&g= t;owner-chemistry ~~ ccl.net</a>> wrote:<br>
</div>
<blockquote class=3D"gmail_quote" style=3D"margin:0px 0px= 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">= ;
<br>
Sent to CCL by: "Stefan  Grimme" [grimme..<a = href=3D"http://thch.uni-bonn.de" rel=3D"noreferrer" = target=3D"_blank">thch.uni-bonn.de</a>]<br>
One more comment to the Mulliken charge discussion:<br>
even methods without a well-defined basis set limit can be useful<br>=
as already mentioned by Marcel Swart. This holds<br>
for the Mulliken atomic charge partitioning in compact MB/DZ basis sets<= br>
(even TZ is often reasonable). For example the DFTB and GFN-xTB tight-bindi= ng methods are fundamentally based on a Mulliken analysis of the density ma= trix and yield physically very reasonable electrostatic energies. In GFN2-x= TB this also works well up to quadrupole
=C2=A0moments.<br>
Its clear that the Mulliken scheme breaks down for AO basis sets containing= <br>
diffuse components but I really would like to see a differentiated view on = the topic (and not as in a recent general statement of a reviewer something= like "I do not think Mulliken charges are trustworthy").= <br>
Best<br>
Stefan Grimme<br>
<br>
<br>
<br>
-=3D This is automatically added to each message by the mailing script =3D-= <br<br=3Dr<br>
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